Computer Aided Assessment of Mathematics by Chris Sangwin

Author:Chris Sangwin [Sangwin, Chris]
Language: eng
Format: mobi
Published: 0101-01-01T00:00:00+00:00

In this approach the unary minus

commutes with multiplication in a

way natural to establishing the

required form of equivalence.

Notice that this test does not include

laws of indices, so x × x ≠ x 2. Since

we are dealing only with nouns − ×

− does not ‘simplify’ to 1. E.g. − x ×

− x ≠ x × x ≠ x 2. An extra rewrite

rule could be added to achieve this,

which would change the

equivalence classes.

A similar approach is taken with

division. Internally in many CAS,

division is removed by transforming

a/ b to a × b−1.

Equivalence up to commutativity

and associativity of addition and

multiplication of the elementary

operations is a particularly useful

test for checking that an answer is

the ‘same’ and ‘simplified’.

Expression tree equality

This ensures that the two

expressions have the same

representation in the data structure

of the CAS. Literally, we perform a

string match on the internal

representation of the data. This is

the strictest notion of all. For

example, the expressions x + y and

y + x have different representations

as trees, but in few situations would

a teacher accept one but not the

other. In practice this test is rarely

helpful, which is why string

matching and regular expressions

are so limited.

6.7 Forms of elementary

mathematical expression

So far we have concentrated on

comparing two expressions.

However, it is also important that

we can establish that an expression

is in a particular form. For

example, that a quadratic is

factored or in a completed square

form. In practice the most useful

tests are not equality with a

canonical form. This is usually too

strict a test in learning and teaching.

Instead we need to consider and

develop something more subtle. We

shall provide only a few illustrative

examples in this section.

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